THE COUPLING OF HEAVE AND PITCH DUE TO SPEED OF ADVANCE 
are 1-144 sec. and 1-023 sec. respectively. For f= 0-2, 
the coupled periods are 1-151 sec. and 1-017 sec.; while 
for f = 0-5, they are |-181 sec. and 0-991 sec. 
5. To sum up the discussion, it seems that the coupling 
terms are of the form — pM U #% and + qM UA, with 
p and q numerical coefficients approximately between 
unity and one-half. From the numerical examples, we 
may conclude that the alteration in resonance frequencies 
is negligible. It would be of interest to examine forced 
oscillations; for instance, with an impressed heaving 
force the response involves pitching as well as heaving, 
and similarly, with an impressed pitching moment. An 
effect of this sort seems to have been observed by Grim.©) 
It is, of course, possible that even a small coupling effect 
might be magnified into something appreciable at or 
near resonance. However, any satisfactory examination 
of this would involve introducing suitable damping terms 
and that is beyond the scope of the present note, the 
purpose of which was to isolate the coupling effect in 
its simplest form, together with the consequent change 
in the resonance frequencies. 
Appendix 
6. We take the origin O at the centre of the spheroid, 
O x along the axis, O z vertically downwards and O y trans- 
versely. We use spheroidal co-ordinates given by 
x=aeul: y=ae(l — p?)t(C — 1)} cosw; 
a a 
z=ae(l — p*)(C2 — 1)t sinw (1) 
The spheroid is given by € = f) = I/e, and for the sub- 
merged half w ranges from 0 to 7m. The spheroid floats 
half immersed in water, and there is a uniform stream U in 
the negative direction of O x; the solid makes small oscilla- 
tions, in which the heaving velocity at any instant is A upwards, 
and the angular pitching velocity is fb in the positive direction 
round Oy. If ¢ is the velocity potential, we assume the 
condition 0 $/dz =O at the upper surface of the water. 
For small oscillations we assume the condition at the 
immersed surface of the solid to hold at the mean equilibrium 
position; thus, in the subsequent work, we neglect the square 
of the fluid velocity due to the oscillations. 
We take for the velocity potential 
fp =Ux —aeUP, (np) Q (O/Q, (Go) 
—hF, (p, C, w) + pb Fo (u, 6, w) (2) 
with a = > y ALFs(H) Q5 (D coss | 
On ee Q) 
a= Dd Bi P5(H) Q5(D cos. | 
n=0s=0 
The expression (2) satisfies the condition 0 ¢/d z = 0 at the 
upper surface of the water. The first two terms represent 
the spheroid in a uniform stream and give zero normal 
velocity over the immersed surface; hence, as for instance in 
Lamb’s Hydrodynamics, p. 142, we must have 
IF IC =aelo(G— 1) + Pi()sinw | 
dFfd 6 = 4a? e?(@ —1)-+Pl(u)sinw { ~ 
for C= 0; 0 <w<rz. 
(4) 
599 
Putting the expressions (3) in (4) and determining the 
coefficients in the usual way, 
2ael) 2n+1(m—s)! G 
AS a 
i wG =e 2=1 (1 + 5)! Qs () ©) 
2a? e 2n+1(n—s)! Ds : 
BS n 
n 37 (G — 1)t 2 = 1 (n + s)! Q3 (fo) (6) 
with the factors C, D given by 
1 1 
=| Pr (Hw) Py (Ww) d os B= | P) (H) Pr (Hw) d (7) 
=r =i 
It should be noted that in the summations in (4) with (5) 
and (6) terms with s = 0 must be taken with a factor 4. 
Further, s is even throughout, while 1 is even in (5) and is 
odd in (6); this follows from the fact that the integrals in (7) 
are only different from zero under these conditions. 
7. The pressure is given by 
p=gpz+pddf[ot+tpU* —tpq’. (8) 
G =O Pd 5,)? + © P/d s_)? + O P/d 5)? (9) 
On the spheroid the last two terms in (9) come only from 
the last two terms of ¢ in (2) and are of the second order. 
Further, on the spheroid, we have the first two terms of ¢ 
in (2) given by 
aUpfl = € Qi (Lo)/Q (Lo)] =aU(l+k)p. 
where k, is the virtual inertia coefficient for axial motion of 
the spheroid; also, on the spheroid, 
d4/d 5, =[U — pHae(G— pW Pf pe . (I) 
Hence, to first order terms in h and ob, we have on the 
spheroid 
2 l= 
with 
(10) 
q lv U* (l +k)? —2aU(1 + kj) 
(— AIF pw + fd F/d pw] (12) 
If — Z is the upward resultant of the fluid pressures, and 
M is the moment about O y, we have 
-z=|| pnas 
1 
=e (3 — 0] sine da | p PL wan (13} 
0 =I 
M = -| (lz —nx)pdS 
7 1 
=1e@e(G= | sino dw | Phu) du (14) 
0 =i 
8. We shall consider separately the contributions of the 
various terms in the pressure defined by (8) and (12). The 
term gp z gives the hydrostatic vertical force, and moment, 
leading to the usual expressions for the restoring force pro- 
portional to the heaving displacement A, and restoring 
moment proportional to the pitching angle %. Then there 
is a steady vertical force arising from the terms in U, and 
corresponding to the bodily sinkage of a ship in motion. 
We obtain this by using for p in (13) the terms 
